A body of mass m is launched up on a rough inclined plane making an angle of 30° with the horizontal. The coefficient of friction between the body and plane is (√x)/5 if the time of ascent is half of the time of descent. The value of x is ________.

Idea: going up, friction adds to gravity; coming down, it subtracts. The same distance is covered both ways, so the times fix the ratio of the two accelerations.

aᵤₚ=g(sin θ+μ cos θ),

a_down=g(sin θ-μ cos θ).

With the same distance, L=1/2a t² gives aᵤₚtᵤₚ²=a_downt_down². Since tᵤₚ=(t_down)/2,

aᵤₚ=4a_down.

sin θ+μ cos θ=4 sin θ-4μ cos θ

5μ cos θ=3 sin θ, so μ=3/5 tan θ.

At θ=30°, tan 30°=1/(√3):

μ=3/(5√3)=(√3)/5.

Comparing with (√x)/5 gives x=3.

Note μ=0.346 is below tan 30°=0.577, so the body does slide back down — as the question assumes.